1. **State the problem:** We are given the polynomial function $$f(x) = -x^2 (x+4)(x+2)$$ and its expanded form $$-x^4 - 6x^3 - 8x^2$$. 2. **Write the polynomial in standard form:** The polynomial is already expanded and written in standard form as $$-x^4 - 6x^3 - 8x^2$$. 3. **Degree of the polynomial:** The degree is the highest power of $$x$$, which is $$4$$. 4. **Leading coefficient:** The coefficient of the term with the highest degree $$x^4$$ is $$-1$$. 5. **End behavior:** For a polynomial $$ax^n$$ where $$n$$ is even and $$a < 0$$, as $$x \to \infty$$, $$f(x) \to -\infty$$ and as $$x \to -\infty$$, $$f(x) \to -\infty$$. But here, since degree is 4 (even) and leading coefficient is negative, the end behavior is: - As $$x \to \infty$$, $$f(x) \to -\infty$$ - As $$x \to -\infty$$, $$f(x) \to -\infty$$ (Note: The user states $$f(x) \to \infty$$ as $$x \to -\infty$$, which is incorrect for even degree with negative leading coefficient.) 6. **Turning points:** The polynomial has 3 turning points as given. 7. **Approximate coordinates of maxima:** At approximately $$(-3.5, 10)$$ and $$(0, 0)$$. 8. **Approximate coordinates of minimum:** At approximately $$(-1.25, -3)$$. 9. **Intervals where the function increases:** $$(-\infty, -3)$$ and $$(-1.5, 0)$$. 10. **Intervals where the function decreases:** $$(-3, -1.5)$$ and $$(0, \infty)$$. **Final answers:** - Standard form: $$-x^4 - 6x^3 - 8x^2$$ - Degree: $$4$$ - Leading coefficient: $$-1$$ - End behavior: as $$x \to \infty, f(x) \to -\infty$$; as $$x \to -\infty, f(x) \to -\infty$$ - Turning points: $$3$$ - Maxima: $$(-3.5, 10), (0, 0)$$ - Minimum: $$(-1.25, -3)$$ - Increasing intervals: $$(-\infty, -3), (-1.5, 0)$$ - Decreasing intervals: $$(-3, -1.5), (0, \infty)$$